Sixteen Eisenstein series
Using the theory of elliptic functions the authors prove several identities which have as particular cases earlier formulas of Ramanujan. For example: \[ E_{2n}-4\sum_{j=1}^\infty\frac{(-1)^j(2j-1)^{2n}q^{2j-1}}{1-q^{2j-1}}=z^{2n+1}p_n(x),\quad n=0,1,2,3,\ldots, \] where \(E_n\) are the Euler numbers defined by \[ \frac{2e^u}{e^{2u}+1}=\frac{E_nu^n}{n!}, \] \(q=\exp(i\pi\tau)\), \(\text{Im}(\tau)>0\), \(z=z(q)=\left(\sum_{n=-\infty}^\infty q^{n^2}\right)^2\), and \(p_n(x)\) is a polynomial of degree \(n\) in \(x\) with rational coefficients.
scientific article; zbMATH DE number 5636934
scientific article